Unit:
Κατεύθυνση Θεωρητικά ΜαθηματικάLibrary of the School of Science
Author:
Patsalos Konstantinos
Supervisors info:
Απόστολος Γιαννόπουλος, καθηγητής, τμήμα Μαθηματικών, ΕΚΠΑ
Original Title:
Διάφορες προσεγγίσεις στην εικασία του υπερεπιπέδου για τον όγκο των τομών κυρτών σωμάτων
Translated title:
Various approaches to the hyperplane conjecture on the volume of sections of convex bodies
Summary:
The hyperplane conjecture (or slicing conjecture) is an open problem in asymptotic convex geometry. It states that there is a uniform lower bound (not depending on the dimension) for the volume of the intersection of a convex body K of volume 1 with a hyperplane that contains the barycentre of K if the hyperplane is chosen so that this volume is maximized. In this thesis, we first present the necessary background from asymptotic convex geometry and then prove the equivalence between the hyperplane conjecture and the isotropic constant conjecture which states that there is a uniform upper bound for the isotropic constants of all convex bodies.
Passing to the existing results, first we show the upper bound of Bourgain for the isotropic constants. In particular, we explain Dar’s argument. We then show that the more general “hyperplane conjecture for log-concave measures” can be reduced to the hyperplane conjecture for convex bodies. Using Paouris’ deviation inequality, we solve the “isomorphic hyperplane conjecture” and prove Klartag’s upper bound for the isotropic constants. The hyperplane conjecture is also reduced to finding bounds for the volume radius of centroid bodies.
We then focus on an alternative proof of Klartag’s upper bound through a variant of the isotropic conjecture. The proof is based on convex cones of n+1 dimensions. Finally, we discuss the best known (till now) upper bound, which is due to Y.Chen and is achieved by stochastic methods.
Main subject category:
Science
Keywords:
Hyperplane, volume, isotropic constant, convex body, log-concave measure, centroid body, convex cone
